Question

The interval in which x must lie so that the greatest term in the expansion of $(1 + x)^{2n}$ has the greatest coefficient is

• A. $\left( \frac{n - 1}{n},\frac{n}{n - 1} \right)$
• B. $\left( \frac{n}{n + 1},\frac{n + 1}{n} \right)$
• C. $\left( \frac{n}{n + 2},\frac{n + 2}{n} \right)$
• D. None of these

Answer 1

Answer: (B)

$\left( \frac{n}{n + 1},\frac{n + 1}{n} \right)$

Answer 2

Here the greatest coefficient is $2n ⥂ C_{n}$

$\therefore$ $2n ⥂ C_{n}x^{n} >^{2n} ⥂ C_{n + 1}x^{n - 1}$ ⇒ $x > \frac{n}{n + 1}$ and

$2nC_{n}x^{n} >^{2n} ⥂ C_{n - 1}x^{n + 1} \Rightarrow x < \frac{n + 1}{n}$. Hence the result is (2)